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Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

Use assume......

Mariusz Iwaniuk 1511
June 16 2017
1 0


 

NULL

> 

restart
> 

`assuming`([int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b)], [b > a])

(-a^4+b^4)^(1/2)/(b^2*a)+2^(1/2)*signum(b)*EllipticK((1/2)*2^(1/2))/b-2*2^(1/2)*signum(b)*EllipticE((1/2)*2^(1/2))/b-((1/2)*I)*2^(1/2)*signum(b)*EllipticPi((-a^2+b^2)^(1/2)*signum(b)/b, 1, (1/2)*2^(1/2))/b

 (1)
> 

`assuming`([int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b)], [b > a, b > 0, a > 0])

(-a^4+b^4)^(1/2)/(b^2*a)+(1/2)*2^(1/2)*EllipticF((-a^2+b^2)^(1/2)/b, (1/2)*2^(1/2))/b-2^(1/2)*EllipticE((-a^2+b^2)^(1/2)/b, (1/2)*2^(1/2))/b

 (2)
> 

int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b, AllSolutions)

piecewise(a < b, piecewise(b = 0, infinity, (EllipticK(I)-EllipticE(I))/b)+piecewise(And(0 < b, a < 0), infinity, 0)+piecewise(And(0 < b, a < -b), -(EllipticF(signum(b), I)-EllipticE(signum(b), I))*(b^(1/2)*(-1/b)^(1/2)-(1/b)^(1/2)*(-b)^(1/2))*signum(b)^2/(b^(1/2)*(-b)^(1/2)), 0)-piecewise(a = 0, -infinity, -b = a, -(1/b)^(1/2)*(EllipticF(signum(b), I)-EllipticE(signum(b), I))*signum(b)^2/b^(1/2), -((-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticE(a/abs(b), I)*signum(b)*a*b-(-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticF(a/abs(b), I)*signum(b)*a*b-a^4+b^4)/((-a^4+b^4)^(1/2)*a*abs(b)^2)), b = a, 0, b < a, -piecewise(a = 0, infinity, -b = a, -(-1/b)^(1/2)*(EllipticF(signum(b), I)-EllipticE(signum(b), I))/(-b)^(1/2), -((-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticE(a/abs(b), I)*signum(b)*a*b-(-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticF(a/abs(b), I)*signum(b)*a*b-a^4+b^4)/((-a^4+b^4)^(1/2)*a*abs(b)^2))-piecewise(And(0 < a, b < 0), infinity, 0)-piecewise(And(b < 0, -b < a), -(EllipticF(signum(b), I)-EllipticE(signum(b), I))*(b^(1/2)*(-1/b)^(1/2)-(1/b)^(1/2)*(-b)^(1/2))*signum(b)^2/(b^(1/2)*(-b)^(1/2)), 0)+piecewise(b = 0, -infinity, (EllipticK(I)-EllipticE(I))/b))

 (3)

``


 

Download Integral_ver2.mw

 

Try this...

Mariusz Iwaniuk 1511
June 14 2017
2 2 
sol4 := fsolve([sol1, sol2], {T = 0...0.5, W = 0...30});

 {T = 0.3216117634, W = 29.46435118}

Mathematica says the same what a Maple.

 

maple_solution.mw

Do it this way:...

Mariusz Iwaniuk 1511
June 10 2017
0 2

 

 eval(sum(sum(x^(q-p), p = 0 .. q), q = 0 .. 10), x = 0)

 

or:

 

eval(value(Sum(Sum(x^(q-p), p = 0 .. q), q = 0 .. 10)), x = 0)

In Maple 2017 maybe like this: J := `a...

Mariusz Iwaniuk 1511
June 05 2017
0 0

In Maple 2017 maybe like this:

J := `assuming`([int(ln(t)^n, t = 0 .. x, AllSolutions = true)], [n::posint])

for n=1:

value(eval(J, n = 1))

is:

x*ln(x)-x

 

Only works for alpha 1 and 2- symbolic s...

Mariusz Iwaniuk 1511
June 02 2017
1 6


 

> 

alpha := 1;

1

 (1)
> 

f := proc (x, y) options operator, arrow; x*y-V^alpha*W/(-a*x-b*y+V)^alpha end proc

proc (x, y) options operator, arrow; y*x-V^alpha*W/(-a*x-b*y+V)^alpha end proc

 (2)
> 

sol := solve([diff(f(x, y), x), diff(f(x, y), y)], {x, y})

{x = b*RootOf(4*_Z^3*b^2-4*V*_Z^2*b+V^2*_Z-V*W*a)/a, y = RootOf(4*_Z^3*b^2-4*V*_Z^2*b+V^2*_Z-V*W*a)}

 (3)
> 

allvalues(sol)

{x = b*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b+(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b)/a, y = (1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b+(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b}, {x = b*(-(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b+((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))))/a, y = -(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b+((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)))}, {x = b*(-(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b-((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))))/a, y = -(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b-((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)))}

 (4)
> 

``


 

Download Worksheet.mw

for alpha=3,4,5,6,7......

and for alpha>0 ,alpha in real

numeric calculation see worksheet:

Numeric_solution.mw

 

 

 

QDifferenceEquations packed is only do ...

Mariusz Iwaniuk 1511
May 07 2017
0 0

QDifferenceEquations packed is only do algebraic calculation.

From wikipedia: qgamma = (1-q)^(1-z)*(product((1-q^(n+1))/(1-q^(n+z)), n = 0 .. infinity))

 

qgamma := proc (z, q) options operator, arrow; (1-q)^(1-z)*(product((1-q^(n+1))/(1-q^(n+z)), n = 0 .. 1000)) end proc

z := .8; q := .9

evalf(qgamma(z, q)) =1.156991553

qgamma.mw

eq1 := 1 = abs(1/(I*Pi*z*(1+I*Pi*a))); e...

Mariusz Iwaniuk 1511
May 03 2017
0 1

eq1 := 1 = abs(1/(I*Pi*z*(1+I*Pi*a)));
eq2 := -(5/9)*Pi = argument(1/(I*Pi*z*(1+I*Pi*a)));

fsolve([eq1, eq2], {a = 0 .. 1, z = 0 .. 1})

{a = 0.5612662116e-1, z = 0.3134740437}

PS: Mathematica gives symbolic answer:

 

 

Using int(func,numeric)...

Mariusz Iwaniuk 1511
April 15 2017
1 5

Using int(func(p),p,numeric).

See file attached :)

evalfandintPerformance.mw

evalfandintPerformance1.mw

This can be useful to you....

Mariusz Iwaniuk 1511
February 23 2017
0 1

Ode_nonlinear.mw

 


 

 

 

From maple Help:"Note that isolate ...

Mariusz Iwaniuk 1511
February 10 2017
0 1

From maple Help:"Note that isolate does not perform integration or differentiation to isolate for expr"

Use a collect function like this : collect(yours equation, diff)

Model_Maple.mw

@spalinowy 

With little help Maple can solve....

Mariusz Iwaniuk 1511
February 10 2017
0 4

Simple code to compute the solution? No.

On basis this webpage:

http://www.maplesoft.com/support/help/Maple/view.aspx?path=examples/pdsolve_boundaryconditions

Answer in the file:

Pde_solve.mw

Convert and IntegrationTools is your fri...

Mariusz Iwaniuk 1511
December 23 2016
2 1

Integral_A.mw
Integral_B.mw
 

> 

restart
> 

f := proc (y) options operator, arrow; Pi*(2*ln(y+sqrt(y^2-4))-2*ln(2))^2 end proc

proc (y) options operator, arrow; Pi*(2*ln(y+sqrt(y^2-4))-2*ln(2))^2 end proc

 (1)
> 

Int(f(y), y = 2 .. exp(1)+exp(-1))

Int(Pi*(2*ln(y+(y^2-4)^(1/2))-2*ln(2))^2, y = 2 .. exp(1)+exp(-1))

 (2)
> 

with(IntegrationTools):
> 

simplify(IntegrationTools:-Change(Int(Pi*(2*ln(y+(y^2-4)^(1/2))-2*ln(2))^2, y = 2 .. exp(1)+exp(-1)), y+sqrt(y^2-4) = x, x), size)

2*Pi*(Int((-ln(x)+ln(2))^2*(x^2-4)/x^2, x = 2 .. exp(1)+exp(-1)+(exp(2)-2+exp(-2))^(1/2)))

 (3)
> 

simplify(value(2*Pi*(Int((-ln(x)+ln(2))^2*(x^2-4)/x^2, x = 2 .. exp(1)+exp(-1)+(exp(2)-2+exp(-2))^(1/2)))), symbolic)

4*(((exp(2)+1)*(exp(4)-2*exp(2)+1)^(1/2)+2*exp(2)+exp(4)+1)*ln(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1)^2+(-2*(exp(2)+1)*(ln(2)+2)*(exp(4)-2*exp(2)+1)^(1/2)+(-4*exp(2)-2*exp(4)-2)*ln(2)-4*exp(4)-4)*ln(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1)+((exp(2)+1)*ln(2)^2+(4*exp(2)+4)*ln(2)-4*exp(1)+5*exp(2)+5)*(exp(4)-2*exp(2)+1)^(1/2)+(2*exp(2)+exp(4)+1)*ln(2)^2+(4*exp(4)+4)*ln(2)-4*exp(1)+2*exp(2)-4*exp(3)+5*exp(4)+5)*Pi*exp(-1)/(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1)

 (4)
> 

Digits := 20:
> 

evalf(4*(((exp(2)+1)*(exp(4)-2*exp(2)+1)^(1/2)+2*exp(2)+exp(4)+1)*ln(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1)^2+(-2*(exp(2)+1)*(ln(2)+2)*(exp(4)-2*exp(2)+1)^(1/2)+(-4*exp(2)-2*exp(4)-2)*ln(2)-4*exp(4)-4)*ln(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1)+((exp(2)+1)*ln(2)^2+(4*exp(2)+4)*ln(2)-4*exp(1)+5*exp(2)+5)*(exp(4)-2*exp(2)+1)^(1/2)+(2*exp(2)+exp(4)+1)*ln(2)^2+(4*exp(4)+4)*ln(2)-4*exp(1)+2*exp(2)-4*exp(3)+5*exp(4)+5)*Pi*exp(-1)/(exp(2)+(exp(4)-2*exp(2)+1)^(1/2)+1))

7.0080014290760108080

 (5)
> 

int(f(y), y = 2 .. exp(1)+exp(-1), numeric)

7.0080014290760108047

 (6)
> 

``


 

Download Integral_B.mw

 

> 

restart
> 

f := proc (y) options operator, arrow; Pi*(2*ln(y+sqrt(y^2-4))-2*ln(2))^2 end proc

proc (y) options operator, arrow; Pi*(2*ln(y+sqrt(y^2-4))-2*ln(2))^2 end proc

 (1)
> 

simplify(convert(f(y), arctan), symbolic)

16*Pi*(arctanh((-1+y+(y^2-4)^(1/2))/(y+(y^2-4)^(1/2)+1))-arctanh(1/3))^2

 (2)
> 

simplify(int(16*Pi*(arctanh((-1+y+(y^2-4)^(1/2))/(y+(y^2-4)^(1/2)+1))-arctanh(1/3))^2, y = 2 .. exp(1)+exp(-1)), symbolic)

32*Pi*((((2+2*exp(3)+4*exp(2)+4*exp(4)+exp(1)+exp(5)+2*exp(6))*arctanh(1/3)+(1/2)*exp(1)+exp(6)+(1/2)*exp(5)+1)*(exp(4)-2*exp(2)+1)^(1/2)+(2+exp(5)+2*exp(6)+exp(7)+2*exp(8)+exp(1)+exp(3)+2*exp(2))*arctanh(1/3)+1+(1/2)*exp(1)-(1/2)*exp(3)-(1/2)*exp(5)-exp(6)+(1/2)*exp(7)+exp(8)-exp(2))*arctanh((-(exp(4)-2*exp(2)+1)^(1/2)-exp(2)+exp(1)-1)/((exp(4)-2*exp(2)+1)^(1/2)+exp(2)+exp(1)+1))+((exp(3)+2*exp(4)+(1/2)*exp(5)+exp(6)+2*exp(2)+(1/2)*exp(1)+1)*(exp(4)-2*exp(2)+1)^(1/2)+1+(1/2)*exp(3)+(1/2)*exp(1)+exp(2)+(1/2)*exp(5)+exp(6)+(1/2)*exp(7)+exp(8))*arctanh(((exp(4)-2*exp(2)+1)^(1/2)+exp(2)-exp(1)+1)/((exp(4)-2*exp(2)+1)^(1/2)+exp(2)+exp(1)+1))^2+((exp(3)+2*exp(4)+(1/2)*exp(5)+exp(6)+2*exp(2)+(1/2)*exp(1)+1)*arctanh(1/3)^2+((1/2)*exp(1)+exp(6)+(1/2)*exp(5)+1)*arctanh(1/3)-(3/4)*exp(1)+(1/2)*exp(2)+(1/2)*exp(6)-(1/2)*exp(3)-(3/4)*exp(5)+(1/2)*exp(4)+1/2)*(exp(4)-2*exp(2)+1)^(1/2)+(1+(1/2)*exp(3)+(1/2)*exp(1)+exp(2)+(1/2)*exp(5)+exp(6)+(1/2)*exp(7)+exp(8))*arctanh(1/3)^2+(1+(1/2)*exp(1)-(1/2)*exp(3)-(1/2)*exp(5)-exp(6)+(1/2)*exp(7)+exp(8)-exp(2))*arctanh(1/3)+1/2-(3/4)*exp(1)+(1/4)*exp(3)+(1/4)*exp(5)-(3/4)*exp(7)+(1/2)*exp(8))*exp(-1)/((exp(3)+2*exp(2)+2*exp(4)+exp(1)+2)*(exp(4)-2*exp(2)+1)^(1/2)+exp(1)+exp(5)+2*exp(6)+2)

 (3)
> 

Digits := 20:
> 

evalf(32*Pi*((((2+2*exp(3)+4*exp(2)+4*exp(4)+exp(1)+exp(5)+2*exp(6))*arctanh(1/3)+(1/2)*exp(1)+exp(6)+(1/2)*exp(5)+1)*(exp(4)-2*exp(2)+1)^(1/2)+(2+exp(5)+2*exp(6)+exp(7)+2*exp(8)+exp(1)+exp(3)+2*exp(2))*arctanh(1/3)+1+(1/2)*exp(1)-(1/2)*exp(3)-(1/2)*exp(5)-exp(6)+(1/2)*exp(7)+exp(8)-exp(2))*arctanh((-(exp(4)-2*exp(2)+1)^(1/2)-exp(2)+exp(1)-1)/((exp(4)-2*exp(2)+1)^(1/2)+exp(2)+exp(1)+1))+((exp(3)+2*exp(4)+(1/2)*exp(5)+exp(6)+2*exp(2)+(1/2)*exp(1)+1)*(exp(4)-2*exp(2)+1)^(1/2)+1+(1/2)*exp(3)+(1/2)*exp(1)+exp(2)+(1/2)*exp(5)+exp(6)+(1/2)*exp(7)+exp(8))*arctanh(((exp(4)-2*exp(2)+1)^(1/2)+exp(2)-exp(1)+1)/((exp(4)-2*exp(2)+1)^(1/2)+exp(2)+exp(1)+1))^2+((exp(3)+2*exp(4)+(1/2)*exp(5)+exp(6)+2*exp(2)+(1/2)*exp(1)+1)*arctanh(1/3)^2+((1/2)*exp(1)+exp(6)+(1/2)*exp(5)+1)*arctanh(1/3)-(3/4)*exp(1)+(1/2)*exp(2)+(1/2)*exp(6)-(1/2)*exp(3)-(3/4)*exp(5)+(1/2)*exp(4)+1/2)*(exp(4)-2*exp(2)+1)^(1/2)+(1+(1/2)*exp(3)+(1/2)*exp(1)+exp(2)+(1/2)*exp(5)+exp(6)+(1/2)*exp(7)+exp(8))*arctanh(1/3)^2+(1+(1/2)*exp(1)-(1/2)*exp(3)-(1/2)*exp(5)-exp(6)+(1/2)*exp(7)+exp(8)-exp(2))*arctanh(1/3)+1/2-(3/4)*exp(1)+(1/4)*exp(3)+(1/4)*exp(5)-(3/4)*exp(7)+(1/2)*exp(8))*exp(-1)/((exp(3)+2*exp(2)+2*exp(4)+exp(1)+2)*(exp(4)-2*exp(2)+1)^(1/2)+exp(1)+exp(5)+2*exp(6)+2))

7.0080014290760108096

 (4)
> 

int(f(y), y = 2 .. exp(1)+exp(-1), numeric)

7.0080014290760108047

 (5)
> 

``


 

Download Integral_A.mw

 

From https://www.easycalculation.com/ana...

Mariusz Iwaniuk 1511
December 17 2016
0 0

From https://www.easycalculation.com/analytical/learn-angle-between-two-curves.php

ANGLE.mw

Angle between two curves is 47,56 degree.

 

With help of Mathematica....

Mariusz Iwaniuk 1511
December 13 2016
0 1

With help of Mathematica:

InverseLAP.mw

 

 

Play with vaules...

Mariusz Iwaniuk 1511
December 13 2016
2 1

You must play with values: Digits,ep,maxmesh, initmesh,abserr

to obtain greater accuracy

 

mplprimes.mw

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